Decomposing a graph into expanding subgraphs

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Three examples of our results are the following:

• A classical result of Lipton, Rose and Tarjan from 1979 states that if is a hereditary family of graphs and every graph in has a vertex separator of size , then every graph in has O(n) edges. We construct a hereditary family of graphs with vertex separators of size such that not all graphs in the family have O(n) edges.
• Trevisan and Arora‐Barak‐Steurer have recently shown that given a graph G, one can remove only 1% of its edges to obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are essentially best possible, even when one is allowed to remove 99% of G's edges.
• Sudakov and the second author have recently shown that every graph with average degree d contains an n‐vertex subgraph with average degree at least and vertex expansion . We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1).

The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super‐linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.     

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a ‐design of order n. The existence problem of ‐designs has been completely solved for the graphs for , for , K_{2, 3} and K_{3, 3}. In this paper, I prove that for all , if there exists a ‐design of order N, then there exists a ‐design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.     

Vertex‐disjoint cycles containing specified vertices in a bipartite graph

A minimum degree condition is given for a bipartite graph to contain a 2‐factor each component of which contains a previously specified vertex. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 145–166, 2004     

Excluding a bipartite circle graph from line graphs

We prove that, for a fixed bipartite circle graph H, all line graphs with sufficiently large rank‐width (or clique‐width) must have a pivot‐minor isomorphic to H. To prove this, we introduce graphic delta‐matroids. Graphic delta‐matroids are minors of delta‐matroids of line graphs and they generalize graphic and cographic matroids. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 183–203, 2009     

Decomposition of the complete graph plus a 1‐factor into cycles of equal length

We determine the necessary and sufficient conditions for the existence of a decomposition of the complete graph of even order with a 1‐factor added into cycles of equal length. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 170–207, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10019     

On Rosa-type labelings and cyclic graph decompositions

A labeling (or valuation) of a graph G is an assignment of integers to the vertices of G subject to certain conditions. A hierarchy of graph labelings was introduced by Rosa in the late 1960s. Rosa showed that certain basic labelings of a graph G with n edges yielded cyclic G-decompositions of K _2n+1 while other stricter labelings yielded cyclic G-decompositions of K _2nx+1 for all natural numbers x. Rosa-type labelings are labelings with applications to cyclic graph decompositions. We survey various Rosa-type labelings and summarize some of the related results. (Communicated by Peter Horák)     

On the Decomposition of Graphs into Complete Bipartite Graphs

In a complete bipartite decomposition π of a graph, we consider the number ϑ(v;π) of complete bipartite subgraphs incident with a vertex v. Let ϑ(G)= ϑ(v;π). In this paper the exact values of ϑ(G) for complete graphs and hypercubes and a sharp upper bound on ϑ(G) for planar graphs are provided, respectively. An open problem proposed by P.C. Fishburn and P.L. Hammer is solved as well.     

On perfect Γ‐decompositions of the complete graph

Generalizing the well‐known concept of an i‐perfect cycle system, Pasotti [Pasotti, in press, Australas J Combin] defined a Γ‐decomposition (Γ‐factorization) of a complete graph K_v to be i‐perfect if for every edge [x, y] of K_v there is exactly one block of the decomposition (factor of the factorization) in which x and y have distance i. In particular, a Γ‐decomposition (Γ‐factorization) of K_v that is i‐perfect for any i not exceeding the diameter of a connected graph Γ will be said a Steiner (Kirkman) Γ‐system of order v. In this article we first observe that as a consequence of the deep theory on decompositions of edge‐colored graphs developed by Lamken and Wilson [Lamken and Wilson, 2000, J Combin Theory Ser A 89, 149–200], there are infinitely many values of v for which there exists an i‐perfect Γ‐decomposition of K_v provided that Γ is an i‐equidistance graph, namely a graph such that the number of pairs of vertices at distance i is equal to the number of its edges. Then we give some concrete direct constructions for elementary abelian Steiner Γ‐systems with Γ the wheel on 8 vertices or a circulant graph, and for elementary abelian 2‐perfect cube‐factorizations. We also present some recursive constructions and some results on 2‐transitive Kirkman Γ‐systems. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 197–209, 2009     

最大匹配的路变换图

图G的最大匹配的路变换图NM(G)是这样一个图,它以G的最大匹配为顶点,如果两个最大匹配M_1与M_2的对称差导出的图是一条路(长度没有限制),那么M_1和M_2在NM(G)中相邻.研究了这个变换图的连通性,分别得到了这个变换图是一个完全图或一棵树或一个圈的充要条件.     

双圈图的最大Estrada指数

令Bn^＋表示顶点个数为礼的双圈二部图的集合．考虑了召吉中图依Estrada指数从大到小的排序问题．利用二部图的Estrada指数和最大特征值之间的关系,当n≥8时,得到了Bn^＋中具有最大和次大Estrada指数的图．     

More on the Bipartite Decomposition of Random Graphs

For a graph , let denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that for every graph G , , where is the maximum size of an independent set of G . Erd?s conjectured in the 80s that for almost every graph G equality holds, that is that for the random graph , with high probability, that is with probability that tends to 1 as n tends to infinity. The first author showed that this is slightly false, proving that for most values of n tending to infinity and for , with high probability. We prove a stronger bound: there exists an absolute constant so that with high probability.     

完全等部二分图kn,n的迭线图Lm(kn,n)的谱特征

研究了完全等部二分图Ｋｎ,ｎ的迭线图Ｌ＾ｍ（Ｋｎ,ｎ）的谱特征,证明了当ｎ≥７时,Ｌ＾ｍ（Ｋｎ,ｎ）以谱为特征。     

关于二部图Km1m2-Hm2的升分解

在文献[2]中作者定义了图的一种新分解-升分解(Ascending subgraph Decomposition简记为ASD)，并提出了一个猜想：任意有正数条边的图都可以升分解．本文主要证明了二部图Km1m2-Hm2(m1≥m2)可以升分解，其中Hm2是至多含m2条边的Km1m2的子图．     

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.     

完全3-部图K_(1，10，n)的交叉数

在上世纪五十年代初,Zarankiewicz猜想完全2-部图Km,m(m≤n)的交叉数为[m/2][m-1/2][n/2][n-1/2](对任意实数x,[x]表示不超过x的最大整数),目前只证明了当m ≤ 6时,Zarankiewicz猜想是正确的.假定Zarankiewicz猜想对m=11的情形成立,本文确定完全3-部图K1,10,n的交叉数.     

The Ihara zeta function of the complement of a semiregular bipartite graph

We study the Ihara zeta function of the complement of a semiregular bipartite graph. A factorization formula for the Ihara zeta function is derived via which the number of spanning trees is computed. For a class of complements of semiregular bipartite graphs, it is shown that they have the same Ihara zeta function if and only if they are cospectral.